A198761 -id:A198761 - cp's OEIS frontend

A225823 Number of hopping sequences in four-colored rooted trees with n nodes, starting and ending with the same "initial state" from all of the (two-colored) rooted trees in A198760. Compared to A198761, only one node color of the initial states is mobile on the tree (Falicov-Kimball model).

1, 4, 54, 1568, 80680, 6510624, 761286848, 121944722176, 25668462562560

Offset: 2

Eva Kalinowski, Jul 30 2013

G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012)

Eva Kalinowski and Władysław Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit arXiv:1106.4938, 2011 (Physical Review B, January 2012)
Martin Paech, Numerische und algebraisch-graphentheoretische Algorithmen für korrelierte Quantensysteme, Dissertation, Hannover, 2015 (in German with an abstract in English).

Cf. A000081, A038055, A136793, A198760, A198761.

Term a(10) added by Martin Paech, Sep 02 2013, calculated on a HP Integrity Superdome 2-32s by courtesy of Hewlett-Packard Development Company, L.P.

A240605 Total number of distinct sequences for the number of double occupancy in the underlying Fermion problem (see comment), i.e., the number of distinct hopping sequences (cf. A198761, A225823) in four-colored rooted trees with n nodes, starting and ending with the same coloring in two colors (cf. A198760, corresponding to zero double-occupancy).

Original entry on oeis.org

1, 2, 10, 59, 397, 2878, 21266, 162732, 1253128, 9839212, 77644825, 620377508, 4981522538, 40351448045, 328421827064, 2690586461296, 22139293490054, 183106636176023, 1520309861062921, 12675106437486945, 106033283581264574, 890035798660219755

Offset: 2

Martin Paech, Apr 09 2014

nonn

The sequences of double-occupancy are generated by the operators T_{+U}, T_{-U}, and T_{0} defined in eq. (8) in Phys. Rev. B 85, 045105 (2012), see below.
Also the number of "island altitude-profiles" of length 2n-1, see examples, which satisfy the following requirements:
(1) Every profile starts and ends at sea-level (zero double-occupancies).
(2) The height increases and decreases with every step at most one unit.
(3) The maximum height does not go beyond floor(n/2).
(4) The minimum height does not fall below sea-level.
(5) Sea-level could only be reached after an even number of steps.
(6) For n even, no plateaus exist at maximum height (= n/2).
(7) For n even, two peaks at maximum height have an even distance.

			n = 2
  0 1 0  |->  T_{+U} T_{-U}  |->  /\
n = 3
                                                     __
  0 1 1 1 0  |->  T_{+U} T_{ 0} T_{ 0} T_{-U}  |->  /  \
  0 1 0 1 0  |->  T_{+U} T_{-U} T_{+U} T_{-U}  |->  /\/\
n = 4
                                                                       ____
  0 1 1 1 1 1 0  |->  T_{+U} T_{ 0} T_{ 0} T_{ 0} T_{ 0} T_{-U}  |->  /    \
                                                                       __/\
  0 1 1 1 2 1 0  |->  T_{+U} T_{ 0} T_{ 0} T_{+U} T_{-U} T_{-U}  |->  /    \
                                                                       __
  0 1 1 1 0 1 0  |->  T_{+U} T_{ 0} T_{ 0} T_{-U} T_{+U} T_{-U}  |->  /  \/\
                                                                       _/\_
  0 1 1 2 1 1 0  |->  T_{+U} T_{ 0} T_{+U} T_{-U} T_{ 0} T_{-U}  |->  /    \
                                                                       /\__
  0 1 2 1 1 1 0  |->  T_{+U} T_{+U} T_{-U} T_{ 0} T_{ 0} T_{-U}  |->  /    \
                                                                       /\/\
  0 1 2 1 2 1 0  |->  T_{+U} T_{+U} T_{-U} T_{+U} T_{-U} T_{-U}  |->  /    \
                                                                       /\
  0 1 2 1 0 1 0  |->  T_{+U} T_{+U} T_{-U} T_{-U} T_{+U} T_{-U}  |->  /  \/\
                                                                         __
  0 1 0 1 1 1 0  |->  T_{+U} T_{-U} T_{+U} T_{ 0} T_{ 0} T_{-U}  |->  /\/  \
                                                                         /\
  0 1 0 1 2 1 0  |->  T_{+U} T_{-U} T_{+U} T_{+U} T_{-U} T_{-U}  |->  /\/  \
  0 1 0 1 0 1 0  |->  T_{+U} T_{-U} T_{+U} T_{-U} T_{+U} T_{-U}  |->  /\/\/\

Alois P. Heinz, Table of n, a(n) for n = 2..400
E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit, arXiv:1106.4938 [cond-mat.str-el], 2011 (Physical Review B 85, 045105, Jan 2012)
E. Kalinowski and M. Paech, Table of island altitude-profiles I(n,k) up to order n = 6.
M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012)

Cf. A198760, A198761, A225823.

b:= proc(x, y, m, v, d) option remember; `if`(y>x or y<0 or
       y>m or v and y=m and d=1 or y=0 and irem(x, 2)=1, 0,
      `if`(x=0, 1, `if`(v and y=m or y=0, 0, b(x-1, y, m, v,
      `if`(d=2, 2, 1-d)))+ `if`(y=0 or y=1 and irem(x, 2)=0, 0,
       b(x-1, y-1, m, v, `if`(d=2, `if`(v and y=m, 1, 2), 1-d)))+
       b(x-1, y+1, m, v, `if`(d=2, 2, 1-d))))
    end:
a:= n-> b(2*n-2, 0, iquo(n, 2, 'r'), r=0, 2):
seq(a(n), n=2..30);  # Alois P. Heinz, May 09 2014

b[x_, y_, m_, v_, d_] := b[x, y, m, v, d] = If[y>x || y<0 || y>m || v && y == m && d==1 || y==0 && Mod[x, 2]==1, 0, If[x==0, 1, If[v && y==m || y==0, 0, b[x-1, y, m, v, If[d==2, 2, 1-d]]] + If[y==0 || y==1 && Mod[x, 2]==0, 0, b[x-1, y-1, m, v, If[d==2, If[v && y==m, 1, 2], 1-d]]] + b[x-1, y+1, m, v, If[d==2, 2, 1-d]]]]; a[n_] := b[2*n-2, 0, Quotient[n, 2], Mod[ n, 2]==0, 2]; Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

Terms a(16) and a(17) are calculated on a HP Integrity Superdome 2-16s by courtesy of Hewlett-Packard Development Company, L.P., by Martin Paech, May 08 2014 (The used algorithm generates explicitly all distinct sequences of double-occupancy, i.e. all valid "island altitude-profiles", and counts them.)

a(18)-a(23) from Alois P. Heinz, May 08 2014

A198760 Number of initial spin configurations in two-colored rooted trees with n nodes.

Original entry on oeis.org

2, 8, 32, 136, 596, 2712, 12642, 60234, 291840, 1434184, 7130640, 35807114, 181339236, 925139186, 4750176056, 24528421712, 127294780994, 663591911824, 3473315219722, 18246162722278, 96169600405626, 508413199626078, 2695245063006696, 14324688031734740

Offset: 2

N. J. A. Sloane, Oct 29 2011

nonn

Also the number of two-colored rooted trees that have for a given color of the root at least one nearest neighbor node of the root in the other color. - Martin Paech, Apr 16 2012

G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011

Alois P. Heinz, Table of n, a(n) for n = 2..500
E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit, arXiv:1106.4938, 2011 (Physical Review B, January 2012).
M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012).

Cf. A000081, A038055, A198761, A225823, A245870.

g:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(t*g(i-1$2, 2)+j-1, j)*g(n-i*j, i-1, t), j=0..n/i)))
    end:
a:= n-> 2*(g(n-1$2, 2) -g(n-1$2, 1)):
seq(a(n), n=2..30);  # Alois P. Heinz, May 12 2014

g[n_, i_, t_] := g[n, i, t] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[t*g[i-1, i-1, 2]+j-1, j]*g[n-i*j, i-1, t], {j, 0, n/i}]]]; a[n_] := 2*(g[n-1, n-1, 2] - g[n-1, n-1, 1]) // FullSimplify; Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Nov 25 2014, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.6465426162329497128927135162..., c = 0.29201514711473716704145008728... . - Vaclav Kotesovec, Sep 12 2014

Terms a(8) and a(9) added by Martin Paech, Apr 16 2012
Term a(10) added by Martin Paech, Jul 30 2013
a(11)-a(25) from Alois P. Heinz, May 12 2014

A226584 Numerators of the series expansion of the ground-state energy of the Hubbard model in the limits of strong coupling and infinite dimensions.

Original entry on oeis.org

-1, -1, -19, -593, -23877, -4496245, -1588528613, -12927125815211

Offset: 2

Eva Kalinowski, Jun 12 2013

a(n) is the numerator of the sum of the weighted contributions from the A198761(n) ways of electron hopping on all of the A198760(n) initial spin-configurations, see eq. A1 in Phys. Rev. B 85, 045105 (2012).

			-1/2, -1/2, -19/8, -593/32, -23877/128, -4496245/2048, ... = A226584/A226585

Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.
M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012).

E. Kalinowski and W. Gluza, Evaluation of high-order terms for the Hubbard model in the strong-coupling limit, Phys. Rev. B 85 (4), 045105 (2012), 8 pages.
M. Paech, W. Apel, E. Kalinowski and E. Jeckelmann, Comparison of computer-algebra strong-coupling perturbation theory and dynamical mean-field theory for the Mott-Hubbard insulator in high dimensions, Phys. Rev. B 90 (24), 245147 (2014), 10 pages.

Cf. A226585.

A226585 Denominators of the series expansion of the ground-state energy of the Hubbard model in the limits of strong coupling and infinite dimensions.

Original entry on oeis.org

2, 2, 8, 32, 128, 2048, 55296, 31850496

Offset: 2

Eva Kalinowski, Jun 12 2013

a(n) is the denominator of the sum of the weighted contributions from the A198761(n) ways of electron hopping on all of the A198760(n) initial spin-configurations, see eq. A1 in Phys. Rev. B 85, 045105 (2012).

			-1/2, -1/2, -19/8, -593/32, -23877/128, -4496245/2048, ... = A226584/A226585

Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.
M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012).

E. Kalinowski and W. Gluza, Evaluation of high-order terms for the Hubbard model in the strong-coupling limit, Phys. Rev. B 85 (4), 045105 (2012), 8 pages.
M. Paech, W. Apel, E. Kalinowski and E. Jeckelmann, Comparison of computer-algebra strong-coupling perturbation theory and dynamical mean-field theory for the Mott-Hubbard insulator in high dimensions, Phys. Rev. B 90 (24), 245147 (2014), 10 pages.

Cf. A226584.

cp's OEIS Frontend

A225823 Number of hopping sequences in four-colored rooted trees with n nodes, starting and ending with the same "initial state" from all of the (two-colored) rooted trees in A198760. Compared to A198761, only one node color of the initial states is mobile on the tree (Falicov-Kimball model).

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A198760 Number of initial spin configurations in two-colored rooted trees with n nodes.

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A226584 Numerators of the series expansion of the ground-state energy of the Hubbard model in the limits of strong coupling and infinite dimensions.

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A226585 Denominators of the series expansion of the ground-state energy of the Hubbard model in the limits of strong coupling and infinite dimensions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs